Integrated Math 2 Unit 9: Probability | Notes and practice

Notes

Basics and counting

$\displaystyle P(\text{event}) = \frac{\text{favorable}}{\text{total}}$ for equally likely outcomes. $P(\text{not } A) = 1 - P(A)$.
Multiplication principle: multiply the number of ways for each stage. Permutations count ordered arrangements; combinations $_nC_r$ count unordered groups.

Compound events

Independent: $P(A \text{ and } B) = P(A)P(B)$. Dependent (for example, without replacement): $P(A \text{ and } B) = P(A)P(B \mid A)$.
Addition rule: $P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$.

Conditional probability and tables

$\displaystyle P(A \mid B) = \frac{P(A \text{ and } B)}{P(B)}$. In a two-way table, divide by the row or column total for the given group.
Events are independent if $P(A \mid B) = P(A)$.

Topics in this unit

Each topic has its own short notes and practice.

Practice

Two fair dice are rolled. What is the probability that the sum is 7?

A teacher picks 2 students from a group of 8 to work together, with no roles. How many pairs are possible?

Events $A$ and $B$ are independent with $P(A) = 0.6$ and $P(B) = 0.3$. What is $P(A \text{ and } B)$?

$P(A) = 0.45$, $P(B) = 0.35$ and $P(A \text{ and } B) = 0.15$. What is $P(A \text{ or } B)$?

$60\%$ of students take Spanish, and $24\%$ take both Spanish and band. What is the probability that a student takes band given that the student takes Spanish?

How many 4-digit PIN codes can be made from the digits 0 to 9 if no digit may repeat?

Free response

Write your answer first, then open the scoring guide.

Free-response question

A bag holds 4 red marbles and 6 blue marbles. Two marbles are drawn one after the other.

(a) Find the probability both are red if the first marble is NOT replaced.

Show the answer and scoring guide

$\displaystyle \frac{4}{10} \cdot \frac{3}{9} = \frac{12}{90} = \frac{2}{15}$.

Check off the points you earned:

1 point: second probability uses 3 red out of 9
1 point: $\frac{2}{15}$

(b) Find the probability both are red if the first marble IS replaced.

Show the answer and scoring guide

The draws are independent: $\displaystyle \frac{4}{10} \cdot \frac{4}{10} = \frac{16}{100} = \frac{4}{25}$.

Check off the points you earned:

1 point: states independence or uses $\frac{4}{10}$ both times
1 point: $\frac{4}{25}$

(c) Without replacement, find the probability that at least one marble is blue.

Show the answer and scoring guide

"At least one blue" is the complement of "both red": $\displaystyle 1 - \frac{2}{15} = \frac{13}{15}$.

Check off the points you earned:

1 point: uses the complement of both red
1 point: $\frac{13}{15}$